Hotel in Osaka

I'm planning a 12-day-trip to Japan and my final destination is Osaka, where I'll be staying for 4 nights then fly back to my country.

To get to Osaka (From Tokyo), I will use JR Pass which will be expired on the first day of my arrival in Osaka. Questions:

1. On the last day I'll have to use local train to get to Kanzai Airport, right? How much does the cheapest one cost?

2. Which area should I stay in order to easily get to both JR and local train station? I'm travelling alone and dragging big luggage.

3. Do you think it's OK if I would go for Chuo Oasis Hotel. It's cheap but I'm not sure in terms of location and safty.

Thank you very much.

Chuou Oasis Horel is located in Nishinari-ku, which is still considered to be "the deep" area of Osaka. It is safer than before, but locals still tend to avoid the area unless you want to visit Tsuukenkaku or enjoy the atmosphere. If you have no problem with Japanese, I don't think you will have any problem.
http://www.chuogroup.jp/oasis/english/access/index.html

I don't know about your budget, but if it is for convenience, I would stay around JR Osaka station area.
http://en.wikipedia.org/wiki/%C5%8Csaka_Station

On the last day I'll have to use local train to get to Kanzai Airport, right?

No, you can also use express trains, buses or taxis. Your options are explained here:
http://www.japan-guide.com/e/e2033.html

Which area should I stay in order to easily get to both JR and local train station?

Osaka/Umeda Station:
http://www.japan-guide.com/e/e4009.html

I agree that the area is "deep", you will see many poor people. And that's why the hotels there, including brand-new Chuo Oasis hotel, are inexpensive. But they never do harm to you.

From JR or Nankai Shinimamiya station to KIX,
890yen by Nankai Express
1030yen by JR Airport Rapid

Thank you very much for every answer.

When visiting the Kansai area I always stay at the Shin Hankyu (spelling?) hotel. You can also get the airpaot shuttle from there. Not the cheapest hotel but I like it as it's in the heart of Osaka. Good Luck!